Symmetry of Second Derivatives¶

Theorem¶

Let $E\subseteq\mathbb{R}^n$ be open, let $\boldsymbol{x}_0\in E$ $E$ , and let $\boldsymbol{f}\colon E\to\mathbb{R}^k$ $R^{k}$ . If the two mixed second-order partial derivatives

\frac{\partial^2 \boldsymbol{f}}{\partial x_j\partial x_i} \quad\text{and}\quad \frac{\partial^2 \boldsymbol{f}}{\partial x_i\partial x_j}

exist on a neighbourhood of $\boldsymbol{x}_0$ and are continuous at $\boldsymbol{x}_0$ , then

\frac{\partial^2 \boldsymbol{f}}{\partial x_j\partial x_i}(\boldsymbol{x}_0) = \frac{\partial^2 \boldsymbol{f}}{\partial x_i\partial x_j}(\boldsymbol{x}_0).

Interpretation¶

When a function is smooth enough, differentiating first in one coordinate direction and then in another gives the same result as doing those two steps in the opposite order.

Commuting square: differentiating f first in x i and then in x j, or in the reverse order, leads to the same mixed second partial.

The order of differentiation does not matter: if both mixed partials exist and are continuous at $\boldsymbol{x}_0$ , they coincide there. Consequence: the Hessian of a $C^2$ function is symmetric. Example: $f=x_1^2x_2$ has $f_{x_1x_2}=f_{x_2x_1}=2x_1$ $2 x_{1}$ .

Twice continuous differentiability is a convenient sufficient hypothesis ensuring the displayed mixed partials are continuous. The stated theorem only needs the specified mixed partials locally; it should not be read as saying every weaker existence assumption implies symmetry.

Key Results¶

A $C^2$ function automatically satisfies the hypotheses of the theorem for second-order mixed partial derivatives.
For scalar-valued $C^2$ functions, the Hessian matrix is symmetric.
The same idea extends to higher-order partial derivatives when the required higher-order derivatives are continuous.

Examples¶

For $f(x,y)=x^2y+\mathrm{e}^{xy}$ $e^{x y}$ , the mixed derivatives $f_{xy}$ and $f_{yx}$ agree at every point because the function is smooth.
For $f(x,y,z)=xyz^2$ $x y z^{2}$ , the derivatives $f_{xz}$ and $f_{zx}$ both equal $2yz$ .

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 1, Lecture 1

Aliases: Schwarz's theorem, Clairaut's theorem, Young's theorem, Schwarz–Clairaut theorem Keywords: equality of mixed partials Symbols: element of, partial, real numbers, subset

Last updated: 17 October 2026 21:53